398
Anal. Chem. 1982, 5 4 , 398-401
First-Order Kinetic Titrimetry Ronald G. Rehm" Center for Applied Mathematics, National Bureau of Standards, Washington, D.C. 20234
David S. Brlght Center for Analytical Chemistty, National Bureau of Standards, Washington, D.C. 20234
We solve the equatlons describing a first-order klnetlc tltratlon by reducing them to a slngle nonlinear ordlnary dlfferential equation. When the rate of addition of tltrant Is constant, the equatlon depends on time and only two parameters, which are related to thls rate and to the equlllbrlum constant. The exact analytlcal solutlon provldes a method for determlnlng the overall character of this chemlcal klnetlcs system and provldes guldellnes for choosing numerical methods to evaluate the solutlon for all values of the parameters. Direct numerlcal lntegratlon of the equation Is convenlent for determlnatlon of titrand concentratlons for some, but nof all condltlons of experlmental Interest. For condltlons approaching those of Ideal tltratlon, Le., of equlllbrlum wlth zero reverse reactlon, numerlcal lntegratlon Is dlfflcult and we supply slmple analytical approxlmatlons of the equlvalence point concentratlon.
Most titrations are performed either in a discrete fashion or with the titrant added sufficiently slowly that the reaction rate between titrand and titrant can be considered to be infinitely fast. This process is known as equilibrium titrimetry. In contrast, a titration process for which the rate of reaction between titrant and titrand is important compared with the rate of addition of titrant is called kinetic titrimetry. In this case, it is important to know reaction rates to infer the original concentration of titrand or the original concentration to infer reaction rates. Such processes have received theoretical consideration in the literature over the past few years (1-4). In previous theoretical treatments, it has been assumed that the forward titration reaction is much faster than the backward reaction so that the equilibrium constant (the ratio of forward to reverse reaction rates) is effectively infinite. It is the purpose of this paper to provide a solution to the kinetic equations under more general conditions when the equilibrium constant is finite. Then the backward reaction rate, as well as the forward reaction rate, becomes a competing process relative to the rate of addition of titrant. Under most conditions of practical interest, the backward reaction rate is very small. However, in some cases it may be important to include this back-reaction or at least to assess its effect on the titration. In ref 4 numerical integration has been used for this purpose, and for a range of values of the rate constants, integration is appropriate. On a more general level, it is of interest to have a complete solution to the kinetic equations describing constant-rate titrations: from this solution the complete character of the process can be ascertained. We first discuss the model and the kinetic equations which describe it. We reduce the equations to a single nonlinear ordinary differential equation in dimensionless form and discuss the significance of the dimensionless parameters. This equation is solved for constant titrant addition rate in terms of parabolic cylinder functions. There are special parametric cases, where simpler analytical expressions can be obtained
and should be used to obtain numerical results. We present time histories of the concentration, regions of validity for different approximate methods, and the concentration at the equivalence point for the complete range of the two parameters which govern this process.
MODEL AND KINETIC EQUATIONS As in ref 1, the chemical reaction is the following: kl
A + B Zk2 C + D
(1)
k = k2/k, where the notation is somewhat different than in that reference: A is the titrant, B the titrand, kl the forward reaction rate, kz the reverse, and k the reciprocal of the equilibrium constant for the reaction. (The reciprocal has been chosen for convenience in the mathematical analysis since k = 0 then represents the special case when the backward reaction is zero.) The kinetic equations describing this reaction are dnA dt
- = -k,nAnB + kzncnD+ p o
(2) (3)
-dnc- - -dnD dt
dnD -d=t - -
dt
dnB dt
(4)
(5)
where ni represents the instantaneous concentration (in mol/L) of chemical i and p o is the constant rate at which titrant A is added (in mol/(L s)). In this formulation it is assumed that the change in concentrations due to addition of titrant is negligible. For the titration problem, the initial conditions are that nA = nc = nD = 0 while nB = ngo at t = 0. Equations 2-5 can be reduced to the following single equation for nB:
As in ref 1,the governing equation (6) is a Ricatti equation. When the reverse rate is neglected (kzis taken to be zero), this equation reduces to eq 6 of ref 1. We note two generalizations of the kinetic system which might be of interest but which are not pursued here. The first concerns more general intitial conditions. If the initial concentrations of A, C , and D are not zero, as assumed above, then the initial concentrations appear as additional constants in eq 6. Since these constants will not change the character of the equation, the procedure for solution-described below-can still be used, and the solution will be given in
This article not subject to US. Copyright. Published 1982 by the American Chemical Society
ANALYTICAL CHEMISTRY, VOL. 54, NO. 3, MARCH 1982
terms of parabolic cylinder functions. The second generalization concerns the rate at which the titrant A is added. If this rate is an arbitrary function of time p ( t ) , then eq 8 still governs the concentration of B with J&p(t? dt’replacing pot. Since this equation if3 still a Ricatti equation, much of the analysis which followe, can be applied directly. In particular, the standard transformation (5) which1 takes this nonlinear first-order equation to a linear second-order equation can be applied. However, at that stage of the analysis, the form of p ( t ) determines the nature of the solution and the ease with which answers can be obtained. We now introduce the dimensionless variables
n
and the definition of a new parameter a =
1 2
k r
- + -(1- K )
(15)
transform eq 11into the standard form for parabolic cylinder functions (6)
g-(: ) -xz+.
w
=o
Two independent solutions to eq 16 are
(7)
nB/nBO 7 I
399
-tPo nBo
and the dimensionless parameters
k
kz/kl
(9)
where M is the confluent hypergeometric function (6),so that the solution is
w = Yl(U,X)
(lo) Here n is the concentration of B normalized by its initial concentration, and T 11~1a dimensionless time. As discussed above, the parameter k is the inverse of the equilibrium constant for the titration reaction in eq 1. The parameter r is the ratio of the rate of addition of A to a rate based upon the forward rate constant for reaction 1 and the initial concentration of B. When r is large, the titration process is characterized by rapid addition of the tiitrant compared with the rate at which the titration reaction proceeds in the forward direction. Equation 6 is now rewritten in dimensionless form using eq 7-10
dn r- = d7
-(T
+ n - l ) n + k ( l - n)2
n=- r 1-k
(a - f / z ) ~ z (a 1 , ~+)CYi(a - Lx)
Y l ( 4
r n - - -1 - k
du/dr u
+ CY2(W)
(19)
To evaluate C, the initial condition must be applied
+
-1 2k n = 1 at t = 0, or a t x = xo = r112
(20)
Then
1-k ---Y1(a,xo) r112
T H E GENERAL SOLUTION The standard transformation (5) which takes the nonlinear, first-order equation (111) into a linear, second-order equation is
(18)
(A second multiplicative constant on the right-hand side of eq 18 would provide the most general solution to the second-order equation (16). However, because of transformation ( I 2 ) ,which requires only the ratio of the derivative to the solution itself, this constant is not needed.) Then, returning to the original dependent variable
(1.1)
Two important pointEi should be made concerning this formulation. First, it can be seen from ecl 11that the form of the solution n is completely determined once the two parameters r and k are known. That is, when the rate of addition of titrant relative to the forward rate constant of the titration and when the equilibrium constant for the titration are specified, the (dimensionless) concentration as a function of (dimensionless) time is known. Carr and Jordan (I) have considered only the special case when k =: 0. Second, a special time of interest is the equivalence point; Le., the time “when the amount of titrant added is stoichiometrically equivalent to n”, as noted in ref 1. This occurs when 7 = 1.
+ CYz(a,x)
C=
- (a - l/z)Yz(a - 1,xo) (1 - k )
(21)
Y l b - 19x0) - r1/2Yz(a,xo) The concentration, ne, at the equivalence point 7 = 1 is of practical interest and can be obtained from this general solution by substituting x = x , = 2k/r1/2into eq 19. For certain values of k and r simple approximations for ne follow. SPECIAL CASES There are three special cases of interest, where simpler expressions for ne can be obtained. The first case, that determined earlier by Carr and Jordan ( I ) , is where the reverse rate constant relative to the forward rate constant can be neglected, k = 0. In the second case k * 1 and by the polynomial approximation 6.1.35 of ref 6 for 0 5 x I 1. (We encountered no problem with the 6 digit calculations although, in general, such limited machine accuracy is known to cause errors (7).)Method r calculates ne from eq 30 and 33. It is the simplest method, yet it works very well when both parameters are small and k < r, i.e., the remainder of the parameter region useful for titration. Figure 1 shows, in particular, the range of the parameters over which approximations denoted by k and r are valid. Where these approximations are valid, we have provided expressions, eq 27 and 28 for method k and eq 30 and 33 for method r, for the concentration at the equivalence point. These expressions can be written explicitly in terms of the titration rate, the initial concentration of titrand, and the forward and backward reaction rates. From the expressions, one can determine the ciensitivity of the equivalence concentration to any of these experimental quantities. We note that when thle parameter r is small, eq 11becomes a singular perturbation problem (see ref 9 for example): the small parameter r multiplies the derivative term, so that when r 0, the differential equation (11)becomes an algebraic equation for n, and the character of the problem changes. In direct numerical integration of eq 11 when r is small, this change of character shows up as a requirement for very small time steps and therefore a very large number of steps to obtain a solution. (The equation becomes stiff in the terminology of numerical analysts.) The large number of time steps produces large cumulativeround-off errora, and the numerical integration can become unsatisfactory. The procedure dascribed in subsection 4(iii) to determine ne utilizes the techniques of singular perturbation theory (9).
-
401
4-3 -5/5
.I
.3
-2
-1 lop
0
I
2
1
Flgure 3. The equivalence point concentration of titrand, ne, as a function of log rand log k . In the lower left, reaction rates for tiation are favorable. When numerical integration succeeds, it generates nB as a function of time. Typical curves are shown in Figure 2 where plots to the lower left represent titration with a favorable reaction (e.g., - 3.8 5 k 5 - 3, - 1 5 log r 5 1 in ref 4) and plots to the top and right represent titrations where little reaction occurs before two stoichiometric amounts of titrant are delivered. The nature of the curves changes with the size of the parameters, suggesting where various data analysis methods might be useful. As expected, straight line extrapolations are indicated in the lower left, whereas numerical curve fitting techniques (as described in ref 4) might be useful where the curves are sensitive to changes in k and r. From eq 6 the slope of the curves is seen to be zero at t = 0. A sharp break in slope at the beginning of a titration indicates small r as seen in the lower plots of Figure 2. (A description of this sharp break can be obtained analytically using singular perturbation techniques; the sharp break will usually not be handled easily by numerical integration.) The behavior of ne as a function of k and r, as computed by the five techniques described above, is shown in Figure 3. From the figure an overall picture can be obtained of the concentration at the equivalence point as a function of the titration rate, the initial concentration of tirand, and the forward and backward reaction rates and how sensitive the concentration is to these experimental quantities. The plot can be used to answer questions such as: (a) How slow must the titration rate be to allow accurate determination of the equilibrium constant or to approach equilibrium conditions? (b) What should the titrand concentration be at the equivalence point, and how sensitive is this concentration to the titration rate? (c) Since r is scaled by the forward rate constant, kl, how will measurement errors of the equivalence concentration affect the value inferred for kl?
LITERATURE CITED (I) Carr, P. W.; Jordan, J. Anal. Chem. 1973, 45, 634. (2) Cover, R. E.; Meltes, L. J . Phys. Chem. 1963, 67, 1528, 2311. (3) Laidler, K. J. ”Chemical Klnetics”; McGraw-Hill: New York, 1965, (4) Campbell, B. H.;Meites, L.; Carr, P. W. Anal. Chem. 1974, 46, 386. (5) Davls, H. T. “Introduction to Nonlinear Differential and Integral Equations”; Dover: New Yok, 1962, Chapter 3. (6) Abramowitz, M., Stegun, I. A., Eds. “Handbook of Mathematical Functions”; US. Government Printing Office: Washington, DC, 1964. (7) Stegun, 1. A.; Zucker, R. J . Res. Natl. Bur. Stand., Sect. B 1970, 745,211. (8) Carnahan, Bruce B.; Luther, H. A.; Wilkes, J. 0. “Appiled Numerlcal Methods“; Wlley: New York, 1969. (9) Lin, C. C.; Segel, L. A. “Mathematics Applied to Deterministic Problems in the Natural Sciences”; Macmiilan: New York, 1974, Chapter 9.
RECEIVED for review July 6,1981. Accepted December 3,1981.
